Also my "10 essential episodes" discussion, part I: TOS through DS9.
Without wanting to reprint the entire spiel from my recovered file, the things to know are:
1) The intent of the list is "if someone who had never watched Star Trek before wanted to get the important points and understanding of the series in as little time as possible going through the whole franchise."
2) I counted multi-part episodes as a single entry. This included non-numbered serialized story continuations like in DS9.
3) I balanced necessity to understand the story of the series (which is why I include pilots even if they're not great), importance to Trek history, and how interesting the episode is itself in terms of important factors in deciding factors for inclusion. I also intentionally avoided anything too goofy (So The Trouble with Tribbles isn't in the TOS list simply because even though it's really fun, it's the kind of campy silliness that might turn someone off if they're on the fence about the franchise)
4) Episodes are listed chronologically, not by order of importance.
5) I also included 5 honorable mentions in case you wanted to expand the list to 15
TOS
1) The Corbomite Maneuver (S1, E2)
2) The Naked Time (S1, E6)
3) Balance of Terror (S1, E8)
4) The Menagerie (S1, E15/16)
5) Space Seed (S1, E24)
6) The Devil in the Dark (S1, E26)
7) The City on the Edge of Forever (S1, E28)
8) The Doomsday Machine (S2, E6)
9) Mirror, Mirror (S2, E10)
10) The Tholian Web (S3, E9)
HM: Where No Man Has Gone Before (S1, E1), Dagger of the Mind (S1, E10), The Galileo Seven (S1, E13), Arena (S1, E19), Journey to Babel (S2, E15)
Comments: The heavy tilt in favor of season 1 wasn't intentional, but turns out to be entirely unsurprising. It makes sense to avoid the mostly disappointing season 2 and the almost-entirely awful season 3. This is also the only time I figure I'll skip out on making the pilot essential viewing mostly because there's little to no continuity between the episodes.
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TNG
1) Encounter at Farpoint (S1, E1/2)
2) Skin of Evil (S1, E23)
3) The Measure of A man (S2, E9)
4) Q Who (S2, E16)
5) Yesterday's Enterprise (S3, E15)
6) The Best of Both Worlds (S3, E26/S4, E1) & Family (S4, E2)*
7)Redemption (S4, E26/S5, E1)
8) I, Borg (S5, E23)
9) Chain of Command (S6, E10/11)
10) Lower Decks (S7, E15)
HM: The Drumhead (S4, E21), Cause & Effect (S5, E18), Relics (S6, E4), The Pegasus (S7, E12), All Good Things... (S7, E25/26)
*While Family isn't strictly part 3 of The Best of Both Worlds, it is for all intents and purposes and I feel it's necessary to close out the storyline.
Comments: If not for their importance, I wouldn't include Encounter at Farpoint or Skin of Evil. Neither is a very good episode. But they each offer something important. Most of the rest of the episodes are no-brainers. Some might question skipping Reunification, but I always felt it was a little dry, and Relics was the better TOS-throwback episode. I actually made a change from my original list, replacing The Pegasus with Lower Decks in the top 10. Lower Decks is such a fantastic, fantastic, fantastic episode. And while I love The Pegasus for its less-than-shiny Starfleet moment and I'm a sucker for a good Terry O'Quinn guest spot, I just had to make the switch because on further reflection Lower Decks is probably in the top 10 Trek episodes ever.
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DS9
1) Emissary (S1, E1/2)
2) Duet (S1, E19)
3) The Maquis (S2, E20/21)
4) The Jem'Hadar (S2, E26) & The Search (S3, E1/2)**
5) The Way of the Warrior (S4, E1/2)
6) In Purgatory's Shadow (S5, E14) & By Inferno's Light (S5, E15)
7) Soldiers of the Empire (S5, E26)
8) Call to Arms (S5, E26)
9) Favor the Bold (S6, E5) & Sacrifice of Angels (S6, E6)
10) In the Pale Moonlight (S6, E19)
11) The Siege of AR-558 (S7, E8)
12) Inter Arma Enim Silent Leges (S7, E16)
HM: Second Skin (S3, E5) and its follow-up episode Ties of Blood And Water (S5, E19), Indiscretion (S4, E5), Hard Time (S4, E19), Far Beyond the Stars (S6, E13), Penumbra through What You Leave Behind (S7, E17-26)
** Like the TNG 3-fer, I'm counting these 3 parts as fitting together into one story.
Comments: Yeah, I cheated big time here, adding 2 episodes to my essentials list. And in total I'm putting like 34 episodes into a list that was meant to be about 15 base. Don't care. Ds9 Is the best Star Trek and its serialized nature makes it necessary to digest in larger chunks. It's worth it. Emissary would be dropped if not for being important setup. Ditto for The Maquis. That'd cut things down to 10 if not for the fact that both episodes inform a lot of what happens in Ds9 and Voyager.
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Maybe tomorrow I'll try to reconstruct my Voyager/Enterprise lists.